Linear Algebra - Sebastian Pancratz

ψ V p = n
ψ(u, v) = −ψ(v, u)
u, v ∈ V ψ(u, u) = 0 u ∈ V A = [ψ]B B V
A T = −A A
A
A = 1
2 (A + AT ) + 1
2 (A − AT )
ψ V
v1, w1, . . . , vm, wm, v2m+1, . . . , vn ψ
⎛
0 1
⎞
0
⎜
⎜−1
⎜
⎝
0
0
−1
1
0
0
⎟ .
⎟
⎠
0 0
dim V = n ψ = 0
v1, w1 ψ(v1, w1) = 1 ψ(w1, v1) = −1 U = 〈v1, w1〉
W = {v ∈ V : ψ(v1, v) = 0 = ψ(w1, v)} V = U ⊕ W v = (av1 + bw1) +
(v − av1 − bw1) a = ψ(v, w1), b = (ψ(v1, v) v − av1 − bw1 ∈ W
U ∩ W = {0} av1 + bw1 ∈ W ψ(av1 + bw1, av1 − bw1) = a 2 + b 2
W
ψ n = 2m
B v1, . . . , vm, w1, . . . , wm, v2m+1, . . . , vn
⎛
0
⎝−Im
Im
0
⎞
0
0⎠
.
0 0 0
U × V U, V
F ψ : U × V → F
U = V ∗ ψ : V ∗ × V → F, (α, v) ↦→ α(v)
F = C ¯ V V V
· λ · v = ¯ λv
¯V × V